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Parameterized Machine Learning for High-Energy Physics
Pierre Baldi, Kyle Cranmer, Taylor Faucett, Peter Sadowski, Daniel Whiteson
TL;DR
High-energy-physics classifiers are commonly trained separately for related tasks such as different particle masses, limiting smooth interpolation across parameter values. This paper adds physics parameters to a neural network’s inputs so one model can address the full task set. The resulting network interpolates to untrained parameter values and can simplify analysis while maintaining or improving performance across the range.
Problem
Related high-energy-physics classification tasks, including searches at different particle masses, are commonly handled by isolated networks that lack smooth interpolation or by single-sample training that sacrifices performance elsewhere.
Method
The paper constructs parameterized neural networks by adding physics parameters θ̄ alongside event-level features x̄, producing outputs f(x̄, θ̄) for different task settings.
Results
A single parameterized network replaces networks trained at individual values, interpolates to untrained cases, and matches a fixed network at an unseen mass without appreciable performance loss.
Takeaways & Limitations
The structure simplifies analyses across theoretical model parameters and can extend to higher-dimensional parameter spaces and nuisance-parameter optimization.
Takeaways & Limitations
Performance depends on the implicit parameter distribution used for training, whose robustness remains an issue.
Abstract
from arXiv · showhide
We investigate a new structure for machine learning classifiers applied to problems in high-energy physics by expanding the inputs to include not only measured features but also physics parameters. The physics parameters represent a smoothly varying learning task, and the resulting parameterized classifier can smoothly interpolate between them and replace sets of classifiers trained at individual values. This simplifies the training process and gives improved performance at intermediate values, even for complex problems requiring deep learning. Applications include tools parameterized in terms of theoretical model parameters, such as the mass of a particle, which allow for a single network to provide improved discrimination across a range of masses. This concept is simple to implement and allows for optimized interpolatable results.
INTRODUCTION
Parameterized neural networks address related high-energy-physics classification tasks by adding physics parameters to the inputs, allowing one network to cover the task set. This can simplify training and support interpolation across parameter values.
- Motivation: Related classification tasks, such as searches across possible particle masses, are often handled by isolated networks that cannot smoothly interpolate.Using one signal sample instead sacrifices performance at other parameter values.
- Approach: A parameterized neural network extends the input features with parameters describing the broader problem, such as a new particle’s mass.The approach applies to classification algorithms, with neural networks providing smooth interpolation.
- Expected benefits: One parameterized network can replace individually trained networks, interpolate to untrained cases, and potentially outperform isolated networks by learning across the parameter range.The paper presents a network structure and examples involving theoretical model parameters with increasing complexity.
NETWORK STRUCTURE & TRAINING
The parameterized network evaluates event features jointly with contextual physics parameters, while training introduces an arbitrary parameter distribution and handling choices for parameters irrelevant to some classes.
- Network structure: A traditional network computes f(x̄), whereas a parameterized network computes f(x̄, θ̄), producing different outputs for different parameter choices.The parameter vector describes the larger context of the learning task.
- Training data: Training examples take the form (x̄, θ̄, y)i, adding the parameter vector and target label to the event-level features.The parameterized inputs let the network learn outputs conditional on both measured features and parameters.
- Training considerations: The training distribution over θ̄ is arbitrary and affects the quality of the resulting network without making the inference Bayesian.The studies use equal-sized samples at a few discrete parameter values.
- Training considerations: Parameters meaningless for a target class are randomly assigned using the same distribution as for the signal class.The examples perform well without tuning, although robustness to the implicit parameter distribution remains an issue.
TOY EXAMPLE
In a toy signal-background problem, a network parameterized by the Gaussian mean learns a response that generalizes between trained parameter values. Its classification accuracy is equally good at trained and untrained values.
- Setup: The toy network uses feature x and parameter θ, with Gaussian signal mean θ, width σ = 0.25, uniform background, and training at θ = −2, −1, 0, 1, 2.Labels 0 and 1 identify background and signal, respectively.
- Interpolation: At θ = −1.5, −0.5, 0.5, and 1.5, the network produces reasonable outputs despite seeing no training examples at those values.The response function retains the same shape as at parameter values represented in training.
- Result: Signal-background classification accuracy is as good at parameter values with no training data as at values represented in training.This supports successful parameterization of the toy solution.
1D PHYSICAL EXAMPLE
The one-dimensional physical study applies parameterization to classifying a hypothetical particle X decaying to t̄t across unknown masses. A network trained at other masses matches a fixed network at an unseen mass, supporting interpolation without appreciable performance loss.
- Physics case: The physical example searches for a hypothetical particle X decaying to t̄t against standard-model t̄t production with the same final state but different kinematics.The signal proceeds through an intermediate resonance, unlike the dominant background.
- Setup: The one-dimensional network uses reconstructed resonance mass mWWbb as its event-level feature, with simulated events passed through showering, hadronization, and detector simulation.The background and signal mass distributions differ across several mX values.
- Baselines: A single network trained at one mass performs best there but degrades elsewhere, while separate mass-specific networks lose continuity away from their training points.Training on an unlabeled mixture reduces some loss away from one mass but smears the signal and degrades nearby performance.
- Parameterized model: The proposed network adds true mass as an input feature, treating n event-level features and m parameters as n+m inputs.Evaluation requires supplying both the event-level features and desired parameter values.
- Evaluation: The interpolation test compares a fixed network trained at mX = 750 GeV with a parameterized network trained at mX = 500, 1000, 1250, 1500 GeV and evaluated at 750 GeV.The parameterized network is tested at a mass absent from its training data.
- Result: The parameterized network’s performance matches that of the fixed network trained at the unseen mass, with no appreciable loss from interpolation.Figure 4 compares the corresponding input distributions and ROC curves.
HIGH-DIMENSIONAL PHYSICAL EXAMPLE
The high-dimensional particle-search example tests whether parameterized networks can learn and interpolate signal-background discrimination across particle masses using low- and high-level event features. At mX = 1000 GeV, interpolation matches a fixed network, while parameterization improves performance across the mass range.
- HIGH-DIMENSIONAL PHYSICAL EXAMPLE: The study tests parameterization and interpolation in a high-dimensional signal-background classification problem.The example searches for a hypothetical particle X decaying to t¯t and uses event-level features together with the mass parameter mX.
- HIGH-DIMENSIONAL PHYSICAL EXAMPLE: The input set contains 21 low-level features from reconstructed event kinematics and five high-level features formed from approximate intermediate-object invariant masses.Low-level inputs include lepton and jet momenta, b-tagging information, and missing transverse momentum.
- HIGH-DIMENSIONAL PHYSICAL EXAMPLE: A parameterized network trained at mX = 500, 750, 1000, 1250, 1500 GeV matches a fixed network trained at mX = 1000 GeV.The networks use seven million training examples, with approximately equal training and testing counts per mass point.
- HIGH-DIMENSIONAL PHYSICAL EXAMPLE: Removing mX = 1000 GeV from training leaves interpolation performance unchanged at mX = 1000 GeV.This demonstrates generalization of the parameterized solution in the high-dimensional example.
- HIGH-DIMENSIONAL PHYSICAL EXAMPLE: Across the mass range, the parameterized network outperforms a network trained only at mX = 1000 GeV, while an unlabeled mixture-trained network has reduced performance at each tested mass.The parameterized network learns the classification problem as a function of mass rather than optimizing only one mass point.
- HIGH-DIMENSIONAL PHYSICAL EXAMPLE: Using only low-level features gives essentially indistinguishable performance for this problem and training-set size.The result is reported for the parameterized deep network without high-level features.
DISCUSSION
The discussion presents parameterized networks as a simplified, more powerful structure for related high-energy-physics tasks. It also identifies extension to higher-dimensional parameter spaces and nuisance-parameter optimization as supported applications.
- DISCUSSION: The paper reports a simplified and more powerful neural-network solution with improved performance across examples of increasing input dimensionality.The demonstrated examples use a single parameter θ.
- DISCUSSION: The technique is easily applied to higher-dimensional parameter spaces.The discussion explicitly extends the single-parameter examples beyond one-dimensional parameterization.
- DISCUSSION: Parameterized networks can optimize performance as a function of nuisance parameters describing systematic uncertainties.Typical networks are optimal only for one nuisance-parameter value used during training.
- DISCUSSION: Profile likelihood ratio tests can select the network corresponding to profiled nuisance-parameter values.This connects parameterized-network outputs to statistical procedures using profile likelihood ratios.